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Organization: Society for Mining, Metallurgy & Exploration

Pages: 10

Publication Date:
Jan 1, 1982

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Abstract

INTRODUCTION Geotechnical models, particularly those based on the finite element method, have been available to aid in en¬gineering design of underground mining excavations for over ten years. Despite this fact there are remarkably few cases of their use in mine design documented in the literature. It is therefore to be anticipated that many potential users of these models are relatively unaware of their capabilities and limitations and also of the form and detail of geotechnical data needed for their success¬ful application. This subsection attempts to address this problem by discussing the different types of numerical models now available and by noting how some of them have been used to study a variety of problems associated with underground mining. The subsection concludes by discussing the applica¬tion of the displacement discontinuity method to the de¬sign of possible mining systems for the copper-nickel deposits of northeastern Minnesota. The object of the analyses, which were nonsite specific, was to determine the significance of geotechnical parameters, such as ini¬tial stress and rock structure, on the stability of under¬ground excavations and hence to provide guidance for future geotechnical investigation. NUMERICAL MODELS In the geomechanical design of underground mining openings, use is made of numerical models that repre¬sent or simulate the large-scale mechanical behavior of rock. There has been less interest in analysis involving fluid flow and heat transfer, but with increasing interest in such areas as in-situ retorting and solution mining it is likely that there will be a growing need for numerical models embracing and coupling all three physical proc¬esses. However, the emphasis in this subsection will be on mechanical behavior. Models which simulate such behavior will be divided into two groups: continuum models and discontinuum models. These will be dis¬cussed in turn in order that some insight into alternative solution strategies and their merits may be gained. Continuum Models Almost all geomechanical numerical models must be classed as continuum models even though particular computer codes incorporate special provisions for rep¬resenting discontinuities such as faults, bedding planes, or joints. They are continuum models because they pro¬vide solutions for cases where material behavior is governed by the differential equations of continuum mechanics. Two basic solution strategies for such equa¬tions may be identified immediately: the differential ap¬proach and the integral approach. In the differential approach a means of approximating the differential equations over the entire region of interest is sought. In the integral approach use is made of fundamental solutions from continuum mechanics, and these are used to construct a solution to the whole problem, making approximations only on the boundaries of the region of interest. The several differential and integral methods are identified in Table 1. Differential Methods: Problems in continuum me¬chanics involve the solution of three types of partial differential equations. Two of these govern the behavior in so-called initial value problems, in which variables change both in time and in space. Examples of such problems include nonsteady heat transfer and fluid flow, and stress wave propagation. The last type of partial differential equation governs the behavior in boundary value problems. In these, variation is in space but not in time. Solution of initial value problems may be achieved in two significantly different methods: implicit and ex¬plicit. The differences between these two methods will be illustrated by considering a very simple initial value problem, that of one-dimensional heat diffusion. The equation governing this process may be written as: [ ] where T is temperature, K is thermal conductivity, p is density, c is specific heat, t is time, and x is the spacial coordinate. In finite difference form this equation might be written as: [ ] where the superscripts refer to the time and the sub¬scripts to the spacial location. Several solution strategies for this equation have been used. Two important ones may be illustrated very simply by discussing the signifi¬cance of the superscript * on the right-hand side of the equation. If i + 1 is substituted for * then the second derivative of the temperature with respect to distance is evaluated at the end of the next time step (using tem¬peratures not already known). Such an approach leads to a set of equations involving unknown temperatures and a solution procedure which is referred to as being implicit. The important characteristic of the implicit procedure is that it leads to a set of equations that must be solved for each time at which the temperature dis¬tribution is required. If instead of substituting (i + 1) for the superscript *, i is substituted, the following equation is obtained: [ ] {In this case the new temperature is defined in terms of an already known temperature distribution. The solution procedure is now known as explicit and has the impor¬tant characteristic that there are no equations that have to be stored or solved. A practical advantage of this |

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